bisection.method {animation} R Documentation

## Demonstration of the Bisection Method for root-finding on an interval

### Description

This is a visual demonstration of finding the root of an equation f(x) = 0 on an interval using the Bisection Method.

### Usage

bisection.method(FUN = function(x) x^2 - 4, rg = c(-1, 10), tol = 0.001,
interact = FALSE, main, xlab, ylab, ...)

### Arguments

 FUN the function in the equation to solve (univariate) rg a vector containing the end-points of the interval to be searched for the root; in a c(a, b) form tol the desired accuracy (convergence tolerance) interact logical; whether choose the end-points by cliking on the curve (for two times) directly? xlab, ylab, main axis and main titles to be used in the plot ... other arguments passed to curve

### Details

Suppose we want to solve the equation f(x) = 0. Given two points a and b such that f(a) and f(b) have opposite signs, we know by the intermediate value theorem that f must have at least one root in the interval [a, b] as long as f is continuous on this interval. The bisection method divides the interval in two by computing c = (a + b) / 2. There are now two possibilities: either f(a) and f(c) have opposite signs, or f(c) and f(b) have opposite signs. The bisection algorithm is then applied recursively to the sub-interval where the sign change occurs.

During the process of searching, the mid-point of subintervals are annotated in the graph by both texts and blue straight lines, and the end-points are denoted in dashed red lines. The root of each iteration is also plotted in the right margin of the graph.

### Value

A list containing

 root the root found by the algorithm value the value of FUN(root) iter number of iterations; if it is equal to ani.options('nmax'), it's quite likely that the root is not reliable because the maximum number of iterations has been reached

### Note

The maximum number of iterations is specified in ani.options('nmax').

Yihui Xie